| L(s) = 1 | + (0.911 + 0.411i)2-s + (0.524 − 1.11i)3-s + (0.660 + 0.750i)4-s + (2.23 − 0.513i)5-s + (0.939 − 0.803i)6-s + (1.07 + 1.75i)7-s + (0.292 + 0.956i)8-s + (0.941 + 1.13i)9-s + (2.24 + 0.450i)10-s + (−0.244 − 1.31i)11-s + (1.18 − 0.345i)12-s + (−3.98 − 0.338i)13-s + (0.261 + 2.04i)14-s + (0.596 − 2.76i)15-s + (−0.127 + 0.991i)16-s + (−4.84 − 1.71i)17-s + ⋯ |
| L(s) = 1 | + (0.644 + 0.291i)2-s + (0.303 − 0.646i)3-s + (0.330 + 0.375i)4-s + (0.997 − 0.229i)5-s + (0.383 − 0.327i)6-s + (0.407 + 0.662i)7-s + (0.103 + 0.338i)8-s + (0.313 + 0.377i)9-s + (0.709 + 0.142i)10-s + (−0.0737 − 0.396i)11-s + (0.342 − 0.0996i)12-s + (−1.10 − 0.0940i)13-s + (0.0698 + 0.545i)14-s + (0.153 − 0.714i)15-s + (−0.0317 + 0.247i)16-s + (−1.17 − 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.55674 + 0.0910244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55674 + 0.0910244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.911 - 0.411i)T \) |
| 223 | \( 1 + (2.98 - 14.6i)T \) |
| good | 3 | \( 1 + (-0.524 + 1.11i)T + (-1.91 - 2.30i)T^{2} \) |
| 5 | \( 1 + (-2.23 + 0.513i)T + (4.49 - 2.18i)T^{2} \) |
| 7 | \( 1 + (-1.07 - 1.75i)T + (-3.15 + 6.25i)T^{2} \) |
| 11 | \( 1 + (0.244 + 1.31i)T + (-10.2 + 3.95i)T^{2} \) |
| 13 | \( 1 + (3.98 + 0.338i)T + (12.8 + 2.19i)T^{2} \) |
| 17 | \( 1 + (4.84 + 1.71i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (0.288 + 0.0163i)T + (18.8 + 2.14i)T^{2} \) |
| 23 | \( 1 + (-1.51 - 1.62i)T + (-1.62 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-7.42 + 3.10i)T + (20.3 - 20.6i)T^{2} \) |
| 31 | \( 1 + (0.326 + 2.08i)T + (-29.5 + 9.49i)T^{2} \) |
| 37 | \( 1 + (6.85 + 5.85i)T + (5.73 + 36.5i)T^{2} \) |
| 41 | \( 1 + (0.157 + 3.71i)T + (-40.8 + 3.47i)T^{2} \) |
| 43 | \( 1 + (-0.578 - 8.16i)T + (-42.5 + 6.06i)T^{2} \) |
| 47 | \( 1 + (8.00 - 0.453i)T + (46.6 - 5.30i)T^{2} \) |
| 53 | \( 1 + (2.98 + 1.78i)T + (25.1 + 46.6i)T^{2} \) |
| 59 | \( 1 + (-0.514 + 0.493i)T + (2.50 - 58.9i)T^{2} \) |
| 61 | \( 1 + (6.26 - 8.98i)T + (-21.1 - 57.2i)T^{2} \) |
| 67 | \( 1 + (3.92 - 0.788i)T + (61.8 - 25.8i)T^{2} \) |
| 71 | \( 1 + (-0.621 - 2.53i)T + (-62.9 + 32.8i)T^{2} \) |
| 73 | \( 1 + (-3.01 + 6.92i)T + (-49.7 - 53.4i)T^{2} \) |
| 79 | \( 1 + (-4.34 + 2.76i)T + (33.5 - 71.5i)T^{2} \) |
| 83 | \( 1 + (0.950 - 13.4i)T + (-82.1 - 11.7i)T^{2} \) |
| 89 | \( 1 + (16.0 + 3.70i)T + (80.0 + 38.9i)T^{2} \) |
| 97 | \( 1 + (5.97 - 6.41i)T + (-6.85 - 96.7i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.32771349508571542356483239014, −10.20054225211489615553505504119, −9.173364577020508159502736047187, −8.270792545004117254162935360079, −7.30782957618080586877272344318, −6.39809526011268432666035530793, −5.36256679145417895018389907273, −4.61154826894989588094414574245, −2.69185526756883328770657058341, −1.91828213306901934457100585461,
1.79610942919716243084891402332, 3.05124055109527001303113394646, 4.41046595562929230452874757075, 4.95280661569576122621652765810, 6.43433693515825022728335968862, 7.07874691110532852757917258932, 8.583615312138161028915812870747, 9.647488800065856843543392453856, 10.22831048261301001600837843436, 10.84701534026092463087813495443
